Problem: $\dfrac{ 2i - 2j }{ -10 } = \dfrac{ i - 4k }{ 7 }$ Solve for $i$.
Multiply both sides by the left denominator. $\dfrac{ 2i - 2j }{ -{10} } = \dfrac{ i - 4k }{ 7 }$ $-{10} \cdot \dfrac{ 2i - 2j }{ -{10} } = -{10} \cdot \dfrac{ i - 4k }{ 7 }$ $2i - 2j = -{10} \cdot \dfrac { i - 4k }{ 7 }$ Multiply both sides by the right denominator. $2i - 2j = -10 \cdot \dfrac{ i - 4k }{ {7} }$ ${7} \cdot \left( 2i - 2j \right) = {7} \cdot -10 \cdot \dfrac{ i - 4k }{ {7} }$ ${7} \cdot \left( 2i - 2j \right) = -10 \cdot \left( i - 4k \right)$ Distribute both sides ${7} \cdot \left( 2i - 2j \right) = -{10} \cdot \left( i - 4k \right)$ ${14}i - {14}j = -{10}i + {40}k$ Combine $i$ terms on the left. ${14i} - 14j = -{10i} + 40k$ ${24i} - 14j = 40k$ Move the $j$ term to the right. $24i - {14j} = 40k$ $24i = 40k + {14j}$ Isolate $i$ by dividing both sides by its coefficient. ${24}i = 40k + 14j$ $i = \dfrac{ 40k + 14j }{ {24} }$ All of these terms are divisible by $2$ $i = \dfrac{ {20}k + {7}j }{ {12} }$